Power functions and homogeneity

**MathBane** · December 4th 2009, 09:13 AM

The question:

Under certain conditions, tsunami waves encountering land will develop into bores. A bore is a surge of water much like what would be expected if a dam failed suddenly and emptied a reservoir into a river bed. In the case of a bore traveling from the ocean into a dry river bed, one study shows that the velocity V of the tip of the bore is proportional to the square root of its height h. This is expressed in the formula below, where k is a constant.

$V = kh^{0.5}$

(a) A bore travels up a dry river bed. How does the velocity of the tip compare with its initial velocity when its height is reduced to one third of its initial height? (Round your answer to two decimal places.)

The velocity changes by a factor of [____].

(b) How does the height of the bore compare with its initial height when the velocity of the tip is reduced to one quarter of its initial velocity?

The height is reduced to [____] of the initial height.

(c) If the tip of one bore surging up a dry river bed is four times the height of another, how do their velocities compare? (Round your answer to two decimal places.)

The velocity of the first is [____] times that of the other.

(I'm not even sure what to do here. How do I use the homogeneity of power functions to help me find the answers when no values are given?)

**HallsofIvy** · December 5th 2009, 04:03 AM

Originally Posted by MathBane

The question:

Under certain conditions, tsunami waves encountering land will develop into bores. A bore is a surge of water much like what would be expected if a dam failed suddenly and emptied a reservoir into a river bed. In the case of a bore traveling from the ocean into a dry river bed, one study shows that the velocity V of the tip of the bore is proportional to the square root of its height h. This is expressed in the formula below, where k is a constant.

$V = kh^{0.5}$

(a) A bore travels up a dry river bed. How does the velocity of the tip compare with its initial velocity when its height is reduced to one third of its initial height? (Round your answer to two decimal places.)

The velocity changes by a factor of [____].

If the initial height is h, then $V_1= kh^{0.5}$ . If the height is reduced to 1/3, then it is h/3 so $V_2= k(h/3)^{0.5}$ . What is the ratio of $V_1$ to $V_2$ ?

(b) How does the height of the bore compare with its initial height when the velocity of the tip is reduced to one quarter of its initial velocity?

The height is reduced to [____] of the initial height.

Now, you are given that $V= kh_1^{0.5}$ and that $V/4= kh_2^{0.5}$ . What is the ratio of those two? What is the ratio of $h_1$ to $h_2$ ?

(c) If the tip of one bore surging up a dry river bed is four times the height of another, how do their velocities compare? (Round your answer to two decimal places.)

The velocity of the first is [____] times that of the other. Really the same as (a) isn't it? $V_1= kh^{0.5}$ and $V_2= k(4h)^{0.5}$ . What is the ratio of $V_2$ to $V_2$ ?

(I'm not even sure what to do here. How do I use the homogeneity of power functions to help me find the answers when no values are given?)

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